A Note on Iterated Consistency and Infinite Proofs

Abstract

Schmerl and Beklemishev's work on iterated reflection achieves two aims: It introduces the important notion of $\Pi^0_1$-ordinal, characterizing the $\Pi^0_1$-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the $\Pi^0_1$-ordinals for a range of theories. The present note demonstrates that these achievements are independent: We read off $\Pi^0_1$-ordinals from a Sch\"utte-style ordinal analysis via infinite proofs, in a direct and transparent way.